\chapter{Results and discussion}
The results from curve fitting experimental data is given in Appendix \ref{sec:curvefits}.

The points $(x,r)=(\frac{3}{8} L, 44) \text{ and } (\frac{3}{8} L,48)$ could not be computed since the for these points $\overline{\xi'^2}>\overline{\xi}$ and thus no pdf could be found with the beta distribution at these points since the coefficients $a$ and $b$ in eq.\ref{eq:beta parameters} are negative.

%Figures
\begin{figure}%
\centering
\includegraphics[height=8cm]{SandiaH2ATemp.png}%
\caption{Distribution of $\tilde{T}$ in K in the Sandia $H_2$-A flame}%
\label{fig:SandiaH2Temp}%
\end{figure}

Fig. \ref{fig:SandiaH2Temp} shows the Favre averaged temperature distribution plotted from the experimental measurements on the Sandia $H_2$-A flame. The measurements where made from the center of the flame and outwards to a maximum radial distance of 56 mm ($r/D=15$) from the centerline. Fig. \ref{fig:SandiaH2mixfrac} shows the distribution of the mixture fraction in the flame.

\begin{figure}%
\centering
\includegraphics[height=10cm]{SandiaH2Amixfrac.jpg}%
\caption{Distribution of $\tilde{\xi}$ in the Sandia $H_2$-A flame}%
\label{fig:SandiaH2mixfrac}%
\end{figure}

\begin{figure}[ht]%
\centering
\subfloat[$\tilde{Y}_{H_2}$]{
	\includegraphics[height=0.4\textheight]{SandiaH2AYH2.jpg}
	\label{fig:SandiaH2YH2}
}
\subfloat[$\tilde{Y}_{O_2}$]{
	\includegraphics[height=0.4\textheight]{SandiaH2AYO2.jpg}
	\label{fig:SandiaH2YO2}
}
\subfloat[$\tilde{Y}_{H_2O}$]{
	\includegraphics[height=0.4\textheight]{SandiaH2AYH2O.jpg}
	\label{fig:SandiaH2YH2O}
}\\
\subfloat[$\tilde{Y}_{OH}$]{
	\includegraphics[height=0.4\textheight]{SandiaH2AYOH.jpg}%
	\label{fig:SandiaH2YOH}
}
\subfloat[$\tilde{Y}_{N_2}$]{
	\includegraphics[height=0.4\textheight]{SandiaH2AYN2.jpg}
	\label{fig:SandiaH2YN2}
}
	
\caption{Distribution of  the Favre averaged species mass fractions in the Sandia $H_2$-A flame.} %{\subref{fig:SandiaH2YH2} is for $\tilde{Y}_{H_2}$,\subref{fig:SandiaH2YO2} is for $\tilde{Y}_{O_2}$,\subref{fig:SandiaH2YH2O} is for $\tilde{Y}_{H_2O}$,\subref{fig:SandiaH2YN2} is for $\tilde{Y}_{N_2}$,\subref{fig:SandiaH2YOH} is for $\tilde{Y}_{OH}$,\subref{fig:SandiaH2YNO} is for $\tilde{Y}_{NO}$}
\label{fig:SandiaH2massfrac}%
\end{figure}

Fig. \ref{fig:SandiaH2massfrac} shows the distribution of the Favre averaged mass fractions of $H_2$, $O_2$, $H_2O$, $N_2$ and $OH$ in the flame.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Results for fast chemistry}
\begin{table}%

\caption{Total rate of entropy production with the fast chemistry assumption}
\begin{tabular}{|l | p{3cm}| p{3cm}| p{3cm} |}
\hline
Source & Mean rate of entropy production, $\dot{\overline{S}}_{gen} [W/K]$ & Irreversibility rate, $\dot{I}=T_{ref}\dot{\overline{S}}_{gen} [W]$ & Percentage of total exergy destruction \\
\hline 
Heat transfer 		& 0.4936 	& 147.1 & 38.51\% \\
Chemical reaction & 0.4796 	& 142.9 & 37.42\% \\
Mass transfer 		& 0.3085	& 91.95	& 24.07\%	\\ \hline
Sum 							& 1.2817  & 381.9 & 100\% \\
\hline
\end{tabular}
\label{tab:fastTotalEntropy}
\end{table}

\begin{figure}%
\includegraphics[width=\columnwidth]{fast_I(x).jpg}%
\label{fig:fast_Sgen(x)}%
\caption{Axial profiles of the rate of entropy production for calculations using the fast chemistry relations in $[W/K]$}%
\end{figure}

\begin{figure}[ht]%
\centering

\subfloat[$(\dot{\overline{S}}_{gen}''')_{Q}$]{
	\includegraphics[height=10cm]{fast_heat.jpg}
	\label{fig:fastheat}
}
\subfloat[$(\dot{\overline{S}}_{gen}''')_{chem}$]{
	\includegraphics[height=10cm]{fast_chem.jpg}
	\label{fig:fastchem}
}
\subfloat[$(\dot{\overline{S}}_{gen}''')_{mass}$]{
	\includegraphics[height=10cm]{fast_mass.jpg}
	\label{fig:fastmass}
}
\label{fig:fastentropyprod}%
\caption{Results of calculations with fast chemistry assumption. Volumetric entropy production in $\frac{W}{K*m^3}$ from \subref{fig:fastheat} heat transfer, \subref{fig:fastchem} chemical reaction, and \subref{fig:fastmass} mass transfer}
\end{figure}


%Sum of entropy production
\begin{figure}[ht]%
\centering

\subfloat[$(\dot{\overline{S}}_{gen})'''_{tot}$]{
	\includegraphics[height=10cm]{fast_sum1.jpg}
	\label{fig:fastsum}
}
\subfloat[$log((\dot{\overline{S}}_{gen})'''_{tot})$]{
	\includegraphics[height=10cm]{fast_sumlog.jpg}
	\label{fig:fastsumlog}
}
\label{fig:fastentropysum}%
\caption{Sum of volumetric entropy production calculated with the fast chemistry assumption. \subref{fig:fastsum} shows $(\dot{\overline{S}}_{gen})'''$ in $\frac{W}{K*m^3}$ and \subref{fig:fastsumlog} is the base 10 logarithm of \subref{fig:fastsum}}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results for curvefits combined with fast chemistry}

\begin{table}%
\caption{Total rate of entropy production with calculations based on curvefits coupled with the fast chemistry relations in Eqs. \ref{eq:Fast fuel mass fraction}-\ref{eq:FastderivT}.}
\begin{tabular}{|l | p{3cm}| p{3cm}| p{3cm} |}
\hline
Source & Mean rate of entropy production, $\dot{\overline{S}}_{gen} [W/K]$ & Irreversibility rate, $\dot{I}=T_{ref}\dot{\overline{S}}_{gen} [W]$ & Percentage of total exergy destruction \\
\hline 
Heat transfer 		& 1.62 	& 484.0 & 6.49\% \\
Chemical reaction & 21.5 & 6416.0 & 85.92\% \\
Mass transfer 		& 1.90	& 566.7	& 7.59\%	\\ \hline
Sum 							& 25.06 & 7467.0 & 100\% \\
\hline
\end{tabular}
\label{tab:curvefitTotalEntropy}
\end{table}

\begin{figure}%
\includegraphics[width=\columnwidth]{curvefit_I(x).jpg}%
\caption{Axial profiles of the rate of entropy production in $[W/m^3K]$ with terms calculated using curvefits coupled with fast chemistry relations}%
\label{fig:curvefit_Sgen(x)}%
\end{figure}

\begin{figure}[ht]%
\centering

\subfloat[$(\dot{\overline{S}}'''_{gen})_{Q}$]{
	\includegraphics[height=10cm]{curvefit_heat.jpg}
	\label{fig:curvefitheat}
}
\subfloat[$(\dot{\overline{S}}'''_{gen})_{chem}$]{
	\includegraphics[height=10cm]{curvefit_chem.jpg}
	\label{fig:curvefitchem}
}
\subfloat[$(\dot{\overline{S}}'''_{gen})_{mass}$]{
	\includegraphics[height=10cm]{curvefit_mass.jpg}
	\label{fig:curvefitmass}
}
\label{fig:curvefitentropyprod}%
\caption{Results of calculations with curvefits. Volumetric entropy production rate in $\frac{W}{K*m^3}$ from \subref{fig:curvefitheat} heat transfer, \subref{fig:curvefitchem} chemical reaction, and \subref{fig:curvefitmass} mass transfer}
\end{figure}

%Sum of entropy production
\begin{figure}[ht]%
\centering

\subfloat[$(\dot{\overline{S}}'''_{gen})_{tot}$]{
	\includegraphics[height=10cm]{curvefit_sum.jpg}
	\label{fig:curvefitsum}
}
\subfloat[$log((\dot{\overline{S}}'''_{gen})_{tot})$]{
	\includegraphics[height=10cm]{curvefit_sumlog.jpg}
	\label{fig:curvefitsumlog}
}
\label{fig:curvefitentropysum}%
\caption{Sum of volumetric entropy production from calculations with curvefits. \subref{fig:curvefitsum} shows $(\dot{\overline{S}}_{gen})'''$ in $\frac{W}{K*m^3}$ and \subref{fig:curvefitsumlog} is the base 10 logarithm of \subref{fig:curvefitsum}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}%
\includegraphics[width=\columnwidth]{velprofiles.jpg}%
\caption{Velocity profiles of the Sandia $H_2$-A flame at different fractions of the of the flame length $L$}%
\label{fig:velprofiles}%
\end{figure}
\section{Possible sources of error}

Some problems occurred when $Y_k(\xi) \rightarrow 0$ since this causes division by zero in the equation for entropy production from mass transfer \ref{eq:massentropy}. This problem was resolved by setting the limits of the integral from $\xi_{st}$ to $1$ for $Y_{H_2}$ and $0$ to $\xi_{st}$ for $Y_{O_2}$. 

Equal diffusivities where assumed and $Le=1$. These assumptions are common practice in combustion engineering and are commonly used for hydrocarbon fuels, but they are not valid for $H_2$-flames. $H_2$ has preferential diffusion and $Le<1$.

The computation time was long for the cells in the outer regions of the flame where the mixture fraction was low. Low values of $\xi$ and $\overline{\xi}$ leads to singularities in the $\beta$-PDF since $a<1$ and $b<1$.  An analytical method of handling these singularities is given in \cite[p.384]{versteeg2007introduction}. This method was not used in the calculations since this only proved to be a problem when calculating the irreversibility contribution of mass transfer. Instead the mass transfer contributions of either $H_2$ or $OH$ was set to zero in the summation term in \emph{Insert reference to mass transfer entropy production!!!!!!!}.


\begin{figure}[h]%
\centering

\subfloat[Fast chemistry]{
	\includegraphics[height=5cm]{IatL1fast.jpg}
	\label{fig:IL1fast}
}
\subfloat[Curvefits]{
	\includegraphics[height=5cm]{IatL1.jpg}
	\label{fig:IL1curve}
}
\label{fig:IL1}%
\caption{Profile of volumetric entropy generation from heat transfer, mass transfer and chemical reaction at $x=1/8 L$. \subref{fig:IL1fast} is from the simulation based on the fast chemistry assumption and \subref{fig:IL1curve} is from the simulation based on the curvefits}
\end{figure}